Circulant Deviation in Structured Matrices

• Circulant deviation is the departure from universal behavior in structured matrices induced by cyclic symmetry.
• It alters statistical distributions and algebraic invariants across random matrix theory, spectral graph theory, and optimization frameworks.
• Applications include predicting non-universal spectral statistics and informing design in coding theory and combinatorial optimization.

Circulant deviation describes the qualitative and quantitative departures from universal, generic, or expected mathematical properties induced by circulant symmetry or patterning, particularly in the contexts of combinatorial, algebraic, probabilistic, and operator-theoretic settings. The concept arises in numerous areas, including random matrix theory, spectral graph theory, combinatorial optimization, and arithmetic invariants, where circulant (or more generally, patterned) structure fundamentally alters statistical distributions, computational invariants, or algebraic behaviors compared to their unstructured or fully generic counterparts.

1. Structural Definition and Overview

A circulant matrix is an $n \times n$ matrix whose rows are successive cyclic right-shifts of its first row, or more generally, a matrix or combinatorial object exhibiting full cyclic (modular) symmetry. In extension, g-circulant matrices generalize this by cyclically shifting by $g$ positions. In many analytic, algebraic, or combinatorial tasks, the imposition of circulant structure induces deviation in the resulting metrics, spectral statistics, or algebraic properties compared to classes with maximal independence or generic structure.

Circulant deviation is observed when the presence of circulant symmetry or associated constraints leads to behavior that substantially diverges from “universal” outcomes—those that are robust under general random or structural assumptions.

2. Circulant Deviation in Random Matrix Statistics

The paper of random matrices often invokes universality: for instance, the Wigner-Dyson statistics for spacings of eigenvalues in the Gaussian Orthogonal Ensemble (GOE). Patterned random matrices with circulant structure strongly violate these universality properties.

For real symmetric circulant and reverse circulant random matrix ensembles with $(n+1)/2$ independent parameters (versus $n(n+1)/2$ for GOE), level spacing statistics demonstrate robust deviation from Wigner's surmise (Ali et al., 2022):

• Symmetric circulant ensemble: Nearest-neighbor level spacings in large $n$ limit obey Poisson (semi-Gaussian) statistics, in stark contrast to the Wigner-Dyson law:

$P(s) = \frac{2}{\pi} e^{-s^2/\pi}$

indicating complete lack of level repulsion.

• Reverse circulant matrices with additional structural zeros: The spacing distribution decays even slower than exponential for large spacings, showing behavior not even conforming to Poisson universality:

$$P(s) = \frac{4A}{\pi}e^{-\frac{4A}{6}s^2}\left[ \int_{0}^\infty \frac{(x+\frac{s}{3})e^{-3Ax^2}}{\sqrt{x(x+\frac{4s}{3})}} dx + \int_{4s/3}^\infty \frac{(x-\frac{s}{3})e^{-3Ax^2}}{\sqrt{x(x-\frac{4s}{3})}} dx\right]$$

(Eq. (29) in (Ali et al., 2022)).

For both ensembles, the reduction in independent entries, when structured as circulant symmetries, drives the deviation. In contrast, Wigner’s universality holds robustly for large, fully symmetric random matrices with no such structure.

3. Deviations in Fluctuations and Spectral Statistics

The fluctuations of linear eigenvalue statistics (such as the trace of powers) for patterned random matrices display further instances of circulant deviation (Adhikari et al., 2016, Maurya et al., 2019):

• CLT for Circulant Ensembles: For random circulant, symmetric circulant, and reverse circulant matrices with i.i.d. standard normal input, the traces of matrix powers (normalized appropriately) converge in total variation distance to a Gaussian, but with variances and normalization exponents dependent on the pattern:

| Matrix Type | CLT Regime | Normalization | Limiting Variance Formula | |------------------------|------------|------------------|-----------------------------------------------| | Circulant | All $p$ | $n^{p+1}$ | $p! \sum_{s=0}^{p-1} f_p(s)$ | | Symmetric Circulant | All $p$ | $n^{p+1}$ | Combinatorial multiline sum | | Reverse Circulant | Even $p$ | $n^{2p+1}$ | $\sum c_k g(k) + 2c_1$ |

The limiting variance for circulant ensembles involves the Irwin-Hall density $f_p(x)$ of the sum of $p$ uniforms, encoding distinct combinatorics driven by the circulant structure.

• Process-level Fluctuations: For circulant matrices with Brownian motion entries, the time-dependent normalized trace processes for different powers $p$ converge to independent Gaussian processes. The covariance structure is diagonal (in $p$), unlike Wigner-type ensembles (Maurya et al., 2019):

$$\mathbb{E}[N_p(t_1)N_q(t_2)] = \delta_{p,q} (\min\{t_1,t_2\})^p p! \sum_{s=0}^{p-1} f_p(s)$$

This separation (i.e., statistical independence between different monomial test functions) is a form of deviation—reflecting the lack of universality in fluctuation structure for patterned/circulant ensembles.

4. Circulant Deviation in Algebraic Structures and Arithmetic Invariants

Circulant deviation manifests in algebraic combinatorics and arithmetic geometry, where the imposition of circulant symmetry restricts or modifies the possible behaviors of invariants such as the $c_2$ arithmetic graph invariant or MDS property of matrices.

• $c_2$ Invariant for Circulant Graphs: For decompleted circulant graphs $\widetilde{C_n}(k_1, k_2)$, the $c_2^{(p)}$ invariant can either be constant, satisfy finite linear recurrences, or vanish, depending strongly on the circulant family and reduction structure (Yeats, 2015). For zigzags,

$$c_2^{(p)}(\widetilde{C_n}(1,2)) = -1 \quad \forall n \geq 5,\ p$$

For $\widetilde{C_n}(1,3)$:

$a_n = a_{n-2} + a_{n-4} + a_{n-6}$

and for large families, explicit finite recurrence systems govern the invariant’s behavior. Only for sufficiently small “gaps” can the method guarantee such recurrences, underscoring a threshold in complexity where circulant structure ceases to enable effective computation.

• MDS Property for Involutory g-Circulant Matrices: In coding theory and matrix analysis, it was conjectured and then established that involutory circulant matrices (i.e., $A^2 = I$) cannot be MDS (maximum distance separable) unless they are left-circulant ($g = -1 \bmod k$). For all other $g$-circulant forms ($g^2 \neq 1 \bmod k$ or $g \neq -1$), structural restrictions induced by circulant deviation preclude the MDS property—any attempt to build such matrices encounters inevitable singular submatrices (Chatterjee et al., 22 Jun 2024).

5. Deviations in Optimization, Polyhedral Theory, and Extremal Structures

Optimization over matrices and polyhedra built from circulant or circular structures demonstrates marked departures in structural and extremal properties:

• Copositive Matrices with Circulant Zero Support: The class of copositive matrices whose zero patterns are circulant admits an explicit semi-definite characterization. Circulant deviation arises in the codimension and extremality of the corresponding faces of the copositive cone: minimal circulant zero support faces form submanifolds of codimension $n$ (extremal only for odd $n$), while non-minimal cases (always extremal) have codimension $2n$ (Hildebrand, 2016). The explicit structure of these faces and their extremality are driven by the interplay between circulant patterning and constraints on zeros, yielding rich arithmetic and algebraic deviation from generic expectations.
• Circulant Minors in Circular Matrices: The classification of contraction minors of circular matrices—a higher-level abstraction of circulant matrices—reveals that the existence of circulant minors is dictated by the presence of families of disjoint directed circuits (with precise winding numbers and no "bad arcs") in associated digraphs (Bianchi et al., 2019). This graph-theoretic condition, induced by circulant structure, governs the occurrence of "non-ideal" submatrices and encodes a deviation from the hereditary universality often present in unpatterned matrices.

6. Quantitative and Qualitative Impact Across Domains

Circulant deviation is not limited to a single area but is manifest across many. In probabilistic limit theorems, the combinatorial structure induced by circulant symmetry impacts not only constants but also the very functional form of fluctuation statistics. In optimization and matrix theory, deviation may translate into codimension, extremal structures, or non-existence results for desirable algebraic properties. In spectral graph theory, circulant deviation quantifies the distance from integrality via algebraic degrees; the minimal order $C(d)$ to realize a given algebraic degree is strictly tied to the circulation parameters and number-theoretic divisibility conditions (Poddar et al., 23 Jul 2025).

Circulant deviation is thus a unified theme characterizing how symmetry and structure restrict or reshape universal phenomena, leading to new classes of behaviors—often simpler, sometimes richer, but always distinct from those of maximally generic or independent objects. These effects are mathematically tractable due to the high degree of symmetry, but simultaneously demonstrate the fragility of universality under strong patterning constraints.

7. Tables: Selected Manifestations of Circulant Deviation

Domain	Circulant Deviation Manifestation	Reference
Random matrix theory	Poisson/semi-Gaussian level spacings in symmetric circulant ensembles	(Ali et al., 2022)
Fluctuations of traces	Non-universal variances, independence across powers	(Adhikari et al., 2016, Maurya et al., 2019)
Arithmetic graph invariants	Recurrence relations/nonexistence of closed forms for $c_2$	(Yeats, 2015)
Matrix algebra/code theory	Non-existence of involutory MDS $g$-circulants except for left-circulant	(Chatterjee et al., 22 Jun 2024)
Copositive cones/optimization	Codimension and extremal face structure in copositive matrices	(Hildebrand, 2016)

8. Conclusion

Circulant deviation represents the class of phenomena where enforcement of circulant symmetry and its relatives produces systematic, often profound departures from universal statistical, algebraic, or combinatorial properties. Its instances span spectral statistics, arithmetic invariants, optimization, and coding theory. The underlying mechanism is the severe reduction of independent degrees of freedom and the concomitant appearance of rigid functional or combinatorial constraints. Understanding and characterizing circulant deviation is crucial for both theoretical insight and practical applications that leverage or contend with patterned structure.